Problem 67 - Intermediate Algebra

Question

Evaluate each expression. See Example \(8 .\) $$ -12 \div 3 \cdot 2 $$

Step-by-Step Solution

Answer

The expression evaluates to -8.

Step 1: Divide First

According to the order of operations, division and multiplication are dealt with from left to right. So first, perform the division part of the expression. Calculate \(-12 \div 3 = -4\)

Step 2: Multiply

Now take the result from Step 1 and multiply it by 2.Calculate\(-4 \cdot 2 = -8\)

Key Concepts

Division and Multiplication

In mathematics, the order of operations determines the sequence in which calculations are performed within an expression. Division and multiplication are on the same precedence level and are executed from left to right. This is crucial to remember because it differs from doing operations as they appear.

• Consider the expression: \(-12 \div 3 \cdot 2\) To solve this, we need to perform operations from left to right, seeing both division and multiplication equally ranked.
• First, solve the division: \(-12 \div 3\). This equals \(-4\).
• Next, multiply the result of the division, \(-4\), by the next number in the expression: \(-4 \cdot 2\).
• The final result of \(-4 \cdot 2\) is \(-8\). This showcases how working from left to right affects the outcome.

Following the correct order is vital for accurate results, especially with problems involving both division and multiplication.

Evaluating Expressions

Evaluating expressions involves simplifying them to find their value. It's important to follow a systematic approach to avoid mistakes.

• Start by identifying each operation in the expression. For our exercise, these are division and multiplication.
• Next, apply the order of operations rules: handle division and multiplication from left to right.
• Substitute and calculate each operation, moving through the expression until simplified.

In the given example, replacing \(-12\) divided by \(3\) with \(-4\), then continuing with \(-4\) times \(2\), demonstrates a successful evaluation process.

• This approach ensures clarity and accuracy while working through mathematical expressions.

Negative Numbers

Working with negative numbers can be tricky, but understanding their properties can simplify calculations.

• A negative divided or multiplied by a positive number remains negative. If you divide \(-12\) by \(3\), the result is \(-4\).
• If you multiply \(-4\) by \(2\), the outcome is \(-8\) because multiplying a negative by a positive number results in a negative.

It's useful to remember:

• Multiplying two negative numbers results in a positive number, but that does not apply to this example as we only involve one negative value.

So, the sequence of operations is emphasized, as handling negative numbers accurately depends on following correct procedures.